5. Intersections of a line segment and a plane

Now let's apply the ideas of the last sections in three dimensions:

(1) The three-point relational form of a plane in    R$^ 3$ is given by the equation $\Delta(P,Q,R,$x$)$ $=$ $0$, where

$\Delta(P,Q,R,$x$) = {\left[\begin{array}{cccc}\ &P&\ &1\\ \ &Q&\ &1\\ \ &R&\ &1\\ \ &\mbox{\bf x}& \ &1\end{array}\right]}$.

(2) $\Delta(P,Q,R,$x$) > 0$ when    x$$ is in the same relation to $P,Q,R$ that the origin is to $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, namely, $P$ to $Q$ to $R$ to $P$ is clockwise as seen from    x$$.

(3) A line segment $\overline{AB}$ crosses the plane given by an affine function $f(x,y,z)$ if $f(A)$ and $f(B)$ are nonzero and have opposite signs; the crossing point $C$ is given by $C$ $=$ $A + t (B-A)$, where $t$ $=$ $- {\frac{\displaystyle f(A)}{\displaystyle f(B) - f(A)}}$.